CHAPTER 2

2.0 Measurement of line parameters It is essential to successfully measure electrical quantities and other parameters in terms of fundamental units, since it determines our knowledge of that quantity. Line parameters are basically: resistance, capacitance and inductance.Other quantities can be derived from these Basic Parameters mainly susceptance, conductance, admittance, e.t.c. other fundamental electrical quantities are voltage and current.

2.1 Measurement of Resistance Resistance may be defined as long medium and high depending on its value and area of application. Generally:i. Low resistance range: 0 - 1 e.g. Shunts, cables, armature, windings e.t.c.ii. Medium Resistance range; 1-105 e.g. common electrical apparatusiii. High Resistance: 103 and above e.g. resistance of insulating materials and the insulation of electrical equipment.

2.2. Measurement of Low Resistances.Three methods are used.Ammeter / Voltmeter MethodPotential meter MethodKelvin Double Bridge Method

2.21 Ammeter / Voltmeter MethodThis measurement is performed by measuring first the voltage across the unknown resistance and then the current. From the ratio v/1 the unknown resistance can be obtained, however the accuracy of this method depends on the accuracy of the instrument used and their relative internal impedance.2.22 Potentiometer MethodThis can be used to measure low resistance provided a standard resistance of a magnitude similar to the unknown and a stable current supply are available. The voltage across the unknown resistance. VA

Is first measured and then the voltage across the known resistance VB is measured. Provided the current magnitude is identical for both measurement.VA = RAI, VB = RBIThus: RA = VA RB VB2.23Kelvin Double BridgeThis method is one of the best available for the precise measurement of low resistance. It eliminates the leads resistance error present in the Wheatstone Bridge

From the figure, x is the load resistance to be measured, s is a standard resistance. X is a low resistance link connecting x and s. Q, q, M, m are four known non-inductive resistances. Either M and m, or Q and q are variable. G is a sensitive detector.AnalysisFor analysis we transform the data arrangement of resistances q, m and r into a star connection below

From the transformation,a = qr q+m+rb = qm q+m+rc = mr q+m+r

The transformed bridgeWhen the bridge is balanced, no current flows through G. therefore, from the Balanced equation of a Wheatstone BridgeQ = x + aM s + c

X + A = Q (s+c) a M X = Q (s+c) a (1) M Substitute a, c in the above equation (1)X = Q S + Q C a M M = Q S + Q mr - qr M M q+m+r q+m+r

X = QS + mr . Q mr . q M q+m+r M q+m+r m

X = QS + mr Q q M q+m+r M m

The termmr Q qq+m+r M m can be made very small by

making the resistance of the link, r very small, and making the ratio Q as close to q as possible m mThis can be achieved easily if m = M, q = QIf this term is made negligibly small, then this expression holds, i.e. x = Q . S M A possible source of error is the generation of thermo e.m.f: To eliminate this, a measurement should also be made with the direction of the current reversed. The mean value of these two readings will give a correct value of x.Range: 1.000 to 100 with an error of 0.1 to 0.01% (accuracy of KDB)

2.3 Medium ResistanceThe method used to measure these are Ammeter / voltmeter methodWheatstone Bridge

2.3.1Ammeter/Voltmeter Method

The accuracy of this method is limited by those of the ammeter and voltmeter used.

Assuming the resistance of the voltmeter (Rv) is very high compared to the unknown resistance (R), then:R = voltmeter reading ammeter readingIf Rv is not much higher than R, a considerable error is introduced in the measurement. A correction can be made as follows.let: R = Unknown Resistance Rm = measure value I = Ammeter Reading or current.Rv is parallel to R; RvllRRm = RRv R + RvRRv = Rm (R+Rv)RRv RmR = RmRvR = RmRv Rv - Rm

2.3.2Wheatstone BridgeThis is best for medium resistance measurements

Simplified Bridge CCT

P and Q are 2 known fixed resistances, s is a variable resistance and R, is unknown resistance G is a sensitive detector S is adjusted until a balance is obtained, this occurs when no current flows through G. at balance, voltage at B equals voltage D.P . V = R . VP+Q R+SP(R+S) = R (P+Q)PR+PS = RP + RQR = P .S QThus, R can be obtained in terms of P, Q and S. P and Q are called the ratio arms. Wheatstone bridge are normally constructed with about 4 or 5 coils of ratio arms which range in tens hundreds and ten thousands(x1, x 10, x 100, x 1000, x 10,000)The variable arm airs then consists of 4 or 5decades of resistance coils.

2.3.2.1Errors in Bridge ArmsThe balance is obtained from the equation. The accuracy measurement which can be obtained with Wheatstone bridge is determined by the errors in the value of the bridge arms.Suppose the arms have small errors+P, +Q, +S.So that error in R = +RR = P x S Q R + R = P + P. (S+ S) Q + QMaximum error can be be obtainedUpper LimitR + R = P + P (S+ S) Q - Q(R+ R) (Q - Q) = (P + P)(S + S)RQ - RQ + QR - RQ= PS + PS + SP + PS

Neglecting 2nd order quantitiesQdR = RQ + PS + SPDividing through by RQ or PS (RQ = PS)QR = RQ +RS +SPRQ RQ PS PSR = Q + S + PR Q S PTherefore, in percentagesI R x 100 = + P x 100 + S x 100 + Q x 100 2.1 R P S QThus, if all the resistances at the given setting of the bridge have small errors. The error in the unknown resistance is the addition of the others in accordance with equation 2.12.4 Measurement of High Resistance The measurement of high resistance is very prone to errors, because parallel leakage paths are difficult to eliminate and the values obtained are affected by

magnitude of the applied voltage the temperature of measurement, the humidity. A high testing voltage is required in this measurement to prove adequate sensitivity.in this measurement, the resistance offered to the flow of current along the surface of insulating is comparable to the resistance to be measured. Among high resistance to be measurements the measurement of insulating resistance of cables in of practical importance and will be described here.The method used for measurement of insulation resistance are:i.Modified Wheatstone bridgeLoss of charge methodThe meggar testerModified Wheatstone bridge

Let, Applied voltage = Vo Resistance of leakage path = RP, Q, r are resistances such thatR>>r>>p>>QWith the bridge at balance and 1-2 open circuit,V1 = I, R, V2 =I2PNow, V1 = V2 I, R = I2PThus R = r ; R=rP P Q QRQ = rPNow if the arm P is unbalanced by a small amount, P, thenThus,V2 = I2 (P+P)V12 = I, R I2 (P+P) I,= I, +I2

And; I2 = R + r .I (R+r) + (P+Q+P)I1 = P+P+Q .I (R+r) + (P+Q+P)V12 = (P+P+Q)R .I (R+r)(P+P) .I (R+r)+(P+P+Q) (R+r)+(P+P+Q)= I (PR + RP + RQ-RP-RP-Pr-rP) (R+r) + (P+P+Q)= I (RQ r (P+P) R+r+P+P+Q)NOTE: QR = rPV12 = I.P.r P+Q+R+r+PTotal resistance of the bridge(R+r) // (P+P+Q)RT = (R+r)(P+P+Q) R+r+P+P+QVo = IRT

I = Vo R+rXP+P+Q P+Q+R+r+PThus:V12 = P.r .V(P+Q+R+r+P) P+Q+R+r+ P (R+r)(P+P+Q)V12 = VoP.r PR+RP+RQ+rP+rP+rQRQ, rP, rP and rQ are much smaller than PR, and therefore can be neglected:V12 = Vo.r.P volts RPThe applied voltage to the detector connected between 1 and 2 is proportional to P/P in arm PThis can be used to measure resistances up to 100G V12 = rP Vo RP

R = rVo P V12 P

Electronic detectors are used to carry out the measurement of such small quantities (as low as O.1pA)

2.4.2Loss of Charge MethodIn this method, the insulation resistance to be measured is connected in parallel with a resistor and electrostatic capacitor. It is charged to a certain voltage, and discharge through the resistor. The terminal voltage is observed over a long period of time by a stop watch.

Net E be the voltage to which the capacitor is charged initially. R is the resistance to be measured. The capacitor terminal voltage. V at any time, t is given byV = Ee-t/RCLog V = log E t / RCt = log E log VRC log E/VR = t Clog e E/VIf the resistance R is very high, the time taken for an appreciate fall in voltage is also large. The voltage vs time (V vs t) graph will be literally flat and this could lead to serious error in calculating and hence R EWhen R is large

More accurate results may be obtained by measuring the change V in the voltage directly.

If this is measured and V = E - VThen the value of R can be obtained from the expression;R = t C log e E E- vThis method can be improved upon by measuring the small voltage drop across R using high gain electronic measuring instrument. The instrument may be calibrated to indicate resistance value directly.

2.4.3The Meggar TesterThese are portable instruments of high resistance measurement. The principles are shown in the figure below.

Principles of Megar Tester (ohmmeter)E and C in the fig. 2 coils fixed at right angle to one another. M is a pivoted magnetic needle to which a pointer is attached. S indicates our supply voltage terminals. X is the resistance to be measured and is connected to coil C. when current flows through C and E their magnetic on M, the magnetic field of E tends to turn the needle in an anti-clockwise direction, while that of C in the clockwise direction. The balance position of The needle is such that these 2 turning moments are equal. If the resistance X is very large ( ) the current in C approaching 0 ( 0) the needle will be set along the axis of coil E. If on the other hand X is very small, the turning moments of C will be much more than that of E and the needle will set along the axis of C. the intermediate points and 0 are obtained by calibration. The scale is graduated in resistance values unusually in mega ohm (MEG) M. Variation of the above are found in commercial testers.

2.4.3.1Meggar Tester for Cable Insulation Resistance MeasurementMeggar is provided with three terminals known as line, earth and guard terminals. The line terminal is connected to the core of the cable, the earth terminal to a plate immersed in the water, or the tank side if suitable and the guard terminal to the guard wire wound tightly round the insulation. Rotating of handle at steady speed then indicates the value of insulation resistance of the cable by final position taken up by the instrument pointer on its scale.

2.5Unbalanced BridgesBridge ccts may be used in a balance or unbalanced conditions. If unbalanced a small change in one of the bridge arms produces a large change in the detector signal, in this way the signal at the galvanometer or the detector may be used to indicate the deviation of an arm from a specifies settings. This is useful in the measurement of dynamic signal in which insufficient time is available to achieve balance conditions.

Bridge CircuitThe voltage across the detector is VAC = E R1 - R2 R1 + R2 R2+R3

The thevenin equivalent of fig. 1 is obtained from figure 2.

RTHfig 2

RTH = R1R4 + R2R3 R1+R4 + R2R3R1 R2

The current through the detector isLg = VACwhere Rg is the RTH +Rgdetector resistanceA modification of this arrangement is commonly used in strain gauge transducers.Quiz 1.If Y = u + vw x Deduce Y from first principle Y Evaluate the max error y and relative error y, whereY u = 5000 + 0.1%v = 600 + 0.5%w = 40 + 1.0% x = 100 + 2.0%

2.The four arms of wheatstone bridge have the following resistances AB, 100, BC 10, CD 4, DA = 50 . A Galvano at 20 is connected across BD. Calculate the current through the galvanometer when a p.d of 10V is measured across AC

2.6 A.C BridgesAlternating current (a.c( circuits are similar to the wheatstone bridge, but they are used to measure inductance and capacitance, in addition to resistance.

In fig. 2.3 the impedance are represented by bridge quantities, Z1, Z2, Z3 and Z4. The principle is to obtain a balanced so that G gives a null reading whenVAB = VADIn both magnitude and phase. Since no current flows through the detector G at balance, then,I1 Z1 = I2Z2i.e. I1 = v z1 + z3

And I2 = V Z2+Z4VZ1 = VZ2Z1+Z3 Z2+Z4

Zi (Z2+Z4) = Z2 (Z 1 + Z2)Z1Z2 + Z1Z4 = Z1Z2 + Z2Z3

Z1 = Z3Z2 Z4

i.e |Z1|

2.6.1 Maxwell Bridge(Measuring Inductance)

L1 is the self inductance to be measured. R1 is the resistance of the inductance R21R3 and R4 are known resistances. C4 is a standard capacitor Balance is obtained by varying C4 and R4 at balance,Z1 = Z2 or Z1 = Z3 Z3 = Z4 Z2 Z4

Z1 = R2 + jwL1 Z2 = R2 Z3 = R4 Z4 = 1 1 R4+ jwC4= R4 I+jwR 4C4R1 + jwL1 = R2 1 + jwc4 R3 R4

= R2 + jwR2C4 R4Equating real parts R1 = R2R3 R4R1 = R2R3 R4

Note that L = jwLC = I jwcj = -1 R = REquating imaginary partsjwL1 = jwR2C4 R3L1 = R2R3C4The imaginary factorsQ = wL / RQ = wL1 = wR2R3C4 R1 = R2R3 / R4 Q = wR4C4W =2f f=50HZW =250

2.6.2Desauty Bridge (Measuring Capacitance)

C1 is the capacitance to be measured. C2 is a standard capacitor. R1 and R2 are non-inductive resistor at balance, R1 1=R21 jwC1 jwC2C1 = R2 C2R1The balance is difficult to obtain in this bridge if the capacitances have dielectric losses.

2.6.3WIEN BRIDGE

C1 is the capacitance to be measured and it has a shunt resistance R1. C2 is a standard air capacitance. R2, R3 and R4 are non-inductive resistors.Balance is obtained by varying R2, R3 and R4at balanceZ1 = Z3Z1Z4 = Z2Z3 Z2 Z4

R3 R2 + = R4 R2R3 + = R4

From the above,C1 = I + W2R22C22R1 = R3 (I + W2 R22C22) W2R2R4C22

NOTE: The heaviside bridges and owen bridges are used for measuring R and L.

2.6.4Schering BridgeThis is used for the measurement of capacitance and dielectric loss, as well as power factors in high voltage networks.jwc21+ jwc111R1R3jwC2R11 + jwC2R4 . C2R3

C1 is the capacitance to be measured C2 is a standard capacitor. C4 is a variable capacitor. R3, R4 are non inductive resistors. R1 represents the dielectric loss of C1.at balance,Z1Z4 = Z2Z3

Now, Z1 = R1 + 1 jwc1Z2 = 1 Z3 = R3 jwc2Z4 = R4 I +jwC4R4R1 + 1 R4 = R3 JwC1 I+jwR4C4 jwC2Equating real termsR1 = C4 . R3 C2 Equating imaginary terms I = R3jwC1 jwC2R4C1 = R4 . C2 R3

The capacitor loss angle is defined astan = WRCFor the circuit above tan = WR1C1= WC4 . R3 . R4 . C2 C2 R3tan = WC4R4

2.7Sources and DetectorsPower is usually supply as alternating current (a.c) or direct current (d . c). For measurement of high frequency the source of a.c. supply is usually an oscillator whereas at power frequencies it is the mains supply. A usual detector used at power frequency is the vibration galvanometer (50 200Hz). At high frequencies (e.g. audio) we can use the headphone (250Hz 4KHZ). The vibration galvanometer is designed for various frequencies but more commonly used below 200HZ a range at which it is very sensitive.

An example of the vibration galvanometer is the moving coil galvanometer. The most versatile of the detectors is the tunable amplifier detector, it is used over a frequency range of (10HZ 100KHZ)

2.8Hall Effect Devices

When a current flows in a conductor which is lar to a magnetic field. A p.d appears between the edges of the material at L angle to the current and the field as illustrated in fig 2.8

The hall voltage VH is more pronounced in semi-conductor materials such as the germanium the voltage is usually small but can be amplified and measured. The hall voltage is given by VH = KIB voltsWhere B is the magnetic field density, K is the hall coefficient of the material for a given current in a material, VH is VHB; I.e hall voltage is proportional to magnetic field density. Instruments based on hall effect can be used to measure constant or slowly varying magnetic field.

2.9Voltmeters & AmmetersCurrent and voltage are commonly measured with ammeters and voltmeters.

Ammeters generally have a low resistance so that when they are connected in series with the cct it doesnt appreciably alter the current. The VM on the otherhand is connected across the pts the p.d. of which it has to be measure. It must therefore have a high resistance so tha the current taken by it is very small. In many applications where the source impedance is large, moving coil and moving iron instrument have a disadvantage. Vms with electronics amplifiers are used in such cases.

fig 2.9

2.9.1Vacuum Tube Voltmeter (VTVM)This can be used for both AC and Dc measurements. It is particularly valuable because of its high input impedance which

Makes it applicable which makes it applicable for measurement of voltage in electronic circuits. Various modification of the arrangement (fig 2.9) are used depending on the range of voltage to be measured. The main advantage of this VM is that it does not load the cct to be measured due to its high impedance. Instead of a vacuum amplifier the instrument may be made of transistor stages with field effect transistor (FET) input stage.

2.9.2Digital VoltmetersA digital voltmeter (DVM) converts a sample of an unknown voltage into a digital quantity which is then display in numerical form. The accuracy with which the input voltage is measured depends largely on the errors in the analogic to digital conversion technic employed

Fig. 2.10Digital Voltmeter